L(s) = 1 | − 4·4-s + 60·11-s + 16·16-s + 278·19-s − 468·29-s − 110·31-s + 276·41-s − 240·44-s + 157·49-s + 792·59-s + 46·61-s − 64·64-s + 408·71-s − 1.11e3·76-s + 1.41e3·79-s + 1.63e3·89-s − 2.55e3·101-s − 1.18e3·109-s + 1.87e3·116-s + 38·121-s + 440·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 1.64·11-s + 1/4·16-s + 3.35·19-s − 2.99·29-s − 0.637·31-s + 1.05·41-s − 0.822·44-s + 0.457·49-s + 1.74·59-s + 0.0965·61-s − 1/8·64-s + 0.681·71-s − 1.67·76-s + 2.01·79-s + 1.94·89-s − 2.51·101-s − 1.04·109-s + 1.49·116-s + 0.0285·121-s + 0.318·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.795004265\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.795004265\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 - 157 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3238 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8062 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 139 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 12530 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 234 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 55 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 64825 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 138 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 156205 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 73690 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 188854 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 396 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 23 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 397222 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 204 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 300553 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 709 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 62030 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 816 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1006321 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.333771564473443982250173851948, −9.198332325488581750882798452470, −8.918711807810749363964605666697, −8.089315436676468617745924990323, −7.80517189333905903915727446340, −7.45412282331189108906789588007, −6.95858885433513371841813469362, −6.79520705165701299812025387110, −5.84408649574187243833065410715, −5.79654734428314153900682027796, −5.19142333856758056245891650680, −5.03118314197903679268832560075, −4.06120804337144436867189475944, −3.84046592406864886065674703122, −3.52749027710271547491704356211, −2.98233355757029309152521709874, −2.14823276783165838146648620826, −1.56476808537157535037200275645, −0.996711207653808070261584069649, −0.54816388286798262476736766119,
0.54816388286798262476736766119, 0.996711207653808070261584069649, 1.56476808537157535037200275645, 2.14823276783165838146648620826, 2.98233355757029309152521709874, 3.52749027710271547491704356211, 3.84046592406864886065674703122, 4.06120804337144436867189475944, 5.03118314197903679268832560075, 5.19142333856758056245891650680, 5.79654734428314153900682027796, 5.84408649574187243833065410715, 6.79520705165701299812025387110, 6.95858885433513371841813469362, 7.45412282331189108906789588007, 7.80517189333905903915727446340, 8.089315436676468617745924990323, 8.918711807810749363964605666697, 9.198332325488581750882798452470, 9.333771564473443982250173851948